where \(u_{i\nu }(\bf r)\)  and \(v_{i\nu }({\bf r})\)  are the electron-like and hole-like wave functions associated with band \(i\)  (\(\nu \)  is the set of the relevant quantum numbers), \(E_{i\nu }\)  represents the quasiparticle energies, and \(\hat{H}_{ei}\)  is the single-particle Hamiltonian (absorbing the chemical potential). For our calculations we assume the effective mass approximation and quasi-2D bands, as emergent multiband superconductors often exhibit quasi-2D Fermi surfaces [1]}, so that
\(\hat{H}_{ei}({\bf r})= -\frac{\hbar ^2}{2m_i}\big (\partial ^2_x + \partial ^2_y\big ) - \mu _i,\) 
where the single-particle energy is degenerate in the \(z\)  direction, \(m_i\)  is the electron band mass, \(\mu _i=m_iv_{Fi}^2/2\)  is the chemical potential measured from the lower edge of the corresponding band, with \(v_{Fi}\)  the band Fermi velocity. The magnetic field is not included as we consider the system in the deep type II regime, where the magnetic field does not vary within the spatial scales of the contributing condensates and cannot influence their spatial lengths.
In addition, for the sake of simplicity we ignore impurities, considering the clean limit.
